Journal of Philosophical Logic

, Volume 42, Issue 5, pp 727–765 | Cite as

Dangerous Reference Graphs and Semantic Paradoxes

Article

Abstract

The semantic paradoxes are often associated with self-reference or referential circularity. Yablo (Analysis 53(4):251–252, 1993), however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook (J Symb Log 69(3):767–774, 2004) and Yablo (2006) with respect to more restricted sentences systems, which we call \(\mathcal{F}\)-systems.

Keywords

Paradox Hypodox Reference structure Circularity Ungroundedness Yablo’s paradox Liar paradox Graph theory Dangerous Precarious \(\mathcal{F}\)-system Kernel 

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References

  1. 1.
    Bruijn, N.G., & Erdos, P. (1951). A colour problem for infinite graphs and a problem in the theory of relations. Indigationes Mathematicae, 13, 371–373.Google Scholar
  2. 2.
    Cook, R.T. (2004). Patterns of paradox. Journal of Symbolic Logic, 69(3), 767–774.CrossRefGoogle Scholar
  3. 3.
    Cook, R.T. (2006). There are non-circular paradoxes (But Yablo’s isn’t one of them!). The Monist, 89(1), 118–149.CrossRefGoogle Scholar
  4. 4.
    Diestel, R. (2010). Graph theory (4th ed.). Heidelberg: Springer.CrossRefGoogle Scholar
  5. 5.
    Eldridge-Smith, P. (2008). The liar paradox and its relatives. PhD thesis, The Australian National University.Google Scholar
  6. 6.
    Herzberger, H. (1970). Paradoxes of grounding in semantics. The Journal of Philosophy, 67(6), 145–167.CrossRefGoogle Scholar
  7. 7.
    Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.CrossRefGoogle Scholar
  8. 8.
    Priest, G. (1997). Yablo’s paradox. Analysis, 57(4), 236–242.CrossRefGoogle Scholar
  9. 9.
    Rabern, L., Rabern, B., Macauley, M. (2012). Dangerous reference graphs and semantic paradoxes. Journal of Philosophical Logic. doi:10.1007/s10992-012-9246-2.Google Scholar
  10. 10.
    Schlenker, P. (2007). The elimination of self-reference: generalized Yablo-series and the theory of truth. Journal of philosophical logic, 36(3), 251–307.CrossRefGoogle Scholar
  11. 11.
    Yablo, S. (1982). Grounding, dependence, and paradox. Journal of Philosophical Logic, 11, 117–137.CrossRefGoogle Scholar
  12. 12.
    Yablo, S. (1985). Truth and reflection. Journal of Philosophical Logic, 14, 297–349.CrossRefGoogle Scholar
  13. 13.
    Yablo, S. (1993). Paradox without self-reference. Analysis, 53(4), 251–252.CrossRefGoogle Scholar
  14. 14.
    Yablo, S. (2006). Circularity and paradox. In S.A. Pedersen, T. Bolander, V.F. Hendricks (Eds.), Self-reference (Chap. 8, pp. 139–157). CSLI Press.Google Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.School of Philosophy, RSSSThe Australian National UniversityCanberraAustralia
  3. 3.Department of Mathematical SciencesClemson UniversityClemsonUSA

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